Transcript No Slide Title

Chapter 4
Graphs of the Circular
Functions
Copyright © 2005 Pearson Education, Inc.
4.1
Graphs of the Sine and
Cosine Functions
Copyright © 2005 Pearson Education, Inc.
Periodic Functions (Conceptual View)


Periodic Functions are functions whose values
repeat in a regular pattern for every member of
their domain
Example:
Copyright © 2005 Pearson Education, Inc.
Slide 4-3
Periodic Functions (Definition)

A “periodic function” is a function f such that
f x  f x  np , for every real number x in the
domain of f, every integer n, and some positive
real number p. The smallest possible positive
value of p is the period of the function.
Whatis thesmallest distancebefore thefunctionbegins to repeat?
2
We say that his
t functionhas a periodof 2.
f x  f x  n2
ALL T RIGINOMETRIC AND CIRCULAR
FUNCT IONSARE PERIODIC
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Slide 4-4
Sine Function

The periodic nature of the circular sine function
can be seen by considering the unit circle and
by graphing the (x, sin x) pairs:
  2      3 2 
0,0,  , ,  ,1,  , ,
4 2 2  4 2 
 5
2   3
 ,0,  , ,  ,1,
2  2

 4
 7
 9 2 
2



,


,

,
2

,
0
,
,
 4
 4 2 
2 



etc.
Copyright © 2005 Pearson Education, Inc.
Slide 4-5
Sine Function f(x) = sin x
Ordered pairs with ratiosshown
as decimalapproximations:

     3

0,0, 
,0.71,  ,1,  ,0.71,
4
2  4

 5
  3
 ,0,  ,0.71,  ,1,
 4
 2

 7

 9



,

0
.
71
,
2

,
0
,
,
0
.
71



,
 4

 4

etc.
2 Period
Periodicwith period2
because : sin x  sin (x  n2 )
Copyright © 2005 Pearson Education, Inc.
Slide 4-6
Notes Concerning the Sine Function



Since the period of the sine function is 2 , all
possible values of the sine function will occur in
any 2 interval, and in each adjoining 2 interval
the exact pattern of values will be repeated
Any 2 interval of values of the sine function is
called “one period” of the sine function
We will define the “primary period” of the sine
function to be those values in the interval 0,2 
Copyright © 2005 Pearson Education, Inc.
Slide 4-7
Sketching the Graph of the Primary Period
of the Sine Function





In a rectangular coordinate system, mark the positions of
0 and 2 on the x-axis
Divide this interval on the x-axis into four intervals and
label the endpoints of each (quarter points)
For each of these five numbers, the corresponding sine
values are 0, 1, 0 -1, 0 (see unit circle)
Plot the (x, sin x) pairs and connect them with a smooth
curve
Graph may be extended left and right
1
1
0

2

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3
2
2
Slide 4-8
Cosine Function

The periodic nature of the circular cosine
function can be seen by considering the unit
circle and by graphing the (x, cos x) pairs:
  2      3
2



0,1,  , ,  ,0 ,  , ,
2 
4 2 2  4
 5
2   3 

 ,1,  , ,  ,0 ,
2  2 
 4
 7 2 
 9 2 




 4 , 2 , 2 ,1,  4 , 2 ,




etc.
Copyright © 2005 Pearson Education, Inc.
Slide 4-9
Cosine Function f(x) = cos x
Ordered pairs with ratiosshown
as decimalapproximations:

     3
0,1,  ,0.71,  ,0 ,  ,0.71,
4
2  4

 ,1,  5 ,0.71,  3 ,0 ,
 4
 2 
 7

 9

,0.71, 2 ,1, 
,0.71,

 4

 4

etc.
2 Period
Periodicwith period2
because : cos x  cos (x  n2 )
Copyright © 2005 Pearson Education, Inc.
Slide 4-10
Notes Concerning the Cosine Function



Since the period of the cosine function is 2 , all
possible values of the cosine function will occur
in any 2 interval, and in each adjoining 2
interval the exact pattern of values will be
repeated
Any 2 interval of values of the cosine function is
called “one period” of the cosine function
We will define the “primary period” of the cosine
function to be those values in the interval 0,2 
Copyright © 2005 Pearson Education, Inc.
Slide 4-11
Sketching the Graph of the Primary Period
of the Cosine Function





In a rectangular coordinate system, mark the positions of
0 and 2 on the x-axis
Divide this interval on the x-axis into four intervals and
label the endpoints of each (quarter points)
For each of these five numbers, the corresponding
cosine values are 1, 0, -1, 0, 1 (see unit circle)
Plot the (x, cos x) pairs and connect them with a smooth
curve
Graph may be extended left and right
1
1
0

2

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3
2
2
Slide 4-12
Amplitude of Sine and Cosine Functions


The amplitude of both the sine and the cosine
function is defined to be “one half of the
difference between the maximum and
minimum values of the function”
Since the maximum value of both functions is “1”
and the minimum value of both is “-1”, the
amplitude of both is:
½[1- (-1)] = ½(2) = 1
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Slide 4-13
Graph of y = a sin x compared with y = sin x





Both graphs will have a period of 2
All the “y” values on y = a sin x, will be multiplied by “a”,
compared with the “y” values on y = sin x
The amplitude of y = a sin x will be:
| a | instead of “1” (the graph will be vertically stretched
or squeezed depending on whether | a | > 1 or | a | < 1)
If a < 0, the graph of y = a sin x will be inverted with
respect to y = sin x
Example: Compare the graphs of:
y = 3 sin x and y = sin x
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Slide 4-14
Example: Compare graph of y = 3 sin x
with graph of y = sin x.

Make a table of values.
Amp 
x
0
/2

3/2
2
sin x
0
1
0
1
0
3sin x
0
3
0
3
0
1
1   1  1 2  1
2
2
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Amp 
1
3   3  1 6  3
2
2
Slide 4-15
Effects of Changing a Factor of the
“Argument” of a Circular Function



In the circular function “sin x”, “x” is called the
argument of the function (The same terminology
applies for each of the other circular functions)
If the argument of a circular function is changed
from “x” to “bx”, the effect is always that:
P
The original period is changed from P to
b
The original graph is horizontally squeezed if
| b | > 1 and horizontally stretched if | b | < 1
These concepts will be discussed and verified
on the following slides
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Slide 4-16
Graph of y = sin x compared with y = sin bx


y = sin x will have primary period for 0  x  2 (has a period of 2 )
y = sin bx will have a primary period for 0  bx  2 , but solving this
inequality for x:
2
0 x
tells us that it has period: b
2
b



Formulafor calculating periodof sin bx
The graph of y = sin bx will have exactly the same shape* as the
graph of y = sin x, except that it will have the period described
above.
In effect, the original graph will be horizontally squeezed or
stretched depending on whether | b | > 1 or | b | < 1
 2 
*The same shape means at the left end of the interval: 0,

it will have a value of 0, at its first quarter point a value  b 
of 1, at its half point a value of 0, at its three quarter point a
value of -1 and at its right end a value of 0
Copyright © 2005 Pearson Education, Inc.
Slide 4-17
Function values for: y = sin x and y = sin 2x
(From Unit Circle or Calculator)
x
0

4

2
3
sin x sin 2x
0
0
.71
1
0
1
.71
1
0
0
 .71
1
x
3
2
7
4
2
sin x sin 2x
1
0
 .71
1
0
0
4

5
4
Copyright © 2005 Pearson Education, Inc.
Slide 4-18
Example: Graph y = sin 2x continued
Period 
Period 2
Period 
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Slide 4-19
Doing a Quick Sketch of y = sin 4x




Primary period of the graph will occur for:

0  4 x  2 Solved for x : 0  x 
(Primaryperiodinterval)
2
Divide this interval into four quarters:
  3 
0, , ,
,
8 4 8 2
Assign to each of these the sine pattern:
0, 1, 0,  1, 0
Graph:
1
1
0
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

8
4
3
8

2
Slide 4-20
Periods of Modified Sine and Cosine
Functions

For b > 0, the graph of y = sin bx will resemble
that of y = sin x, but with period 2 .
b

For b > 0, the graph of y = cos bx will resemble
that of y = cos x, with period 2 .
b
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Slide 4-21
Doing a Quick Sketch of:




2
y  cos x
3
Primary period of the graph will occur for:
2x
0
 2 Solved for x : 0  x  3 (Primaryperiodinterval)
3
Find numbers to divide this interval into four
3 3 9
quarters: 0, , ,
, 3
4 2 4
Assign to each of these the cosine pattern:
1, 0,  1, 0, 1
Graph:
1
1
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3
4
3
2
9
3
4
Slide 4-22
Summary Comments about Graphs of
y = a sin bx and y = a cos bx

Each of these graphs will, respectively, look like
the the graphs of the basic functions, y = sin x
and y = cos x, except that:
“a” alters the amplitude of the graph to | a |
“a” inverts the graph, if “a” is negative
2
“b” , when it is positive, changes the period to:
b
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Slide 4-23
Guidelines for Sketching Graphs of
Sine and Cosine Functions




To graph y = a sin bx or y = a cos bx, with
b > 0,
follow these steps.
Step 1
To find the interval for one period, solve the
inequality: 0  bx  2 (the number at the
right end will be the new period) and lay off
this interval on the x-axis
Step 2
Find numbers to divide the interval into four
equal parts.
Step 3
Assign the appropriate sine or cosine
patterns, multiplied by “a”, to each of these
five x-values. (The points will be maximum
points, minimum points, and x-intercepts.)
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Slide 4-24
Guidelines for Sketching Graphs of
Sine and Cosine Functions continued

Step 4

Step 5
Plot the points found in Step 3, and join
them with a smooth curve having
amplitude |a|.
Draw the graph over additional
periods, to the right and to the left, as
needed.
Copyright © 2005 Pearson Education, Inc.
Slide 4-25
Graph y = 2 sin 4x

Step 1
0  4 x  2
Solve inequality: 0 4 x 2


4 4
4
0 x

Step 2

Step 3

2


 T henew periodis 
2

Divide the interval into four equal parts.
  3 
0, , , ,
8 4 8 2
Assign pattern of sine values (0, 1, 0, -1, 0)
multiplied by “a” (-2) to each of these five
numbers:

      
0, 0,  ,  2 ,  , 0 ,  , 2 ,  , 0 
8
 4  8  8 
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Slide 4-26
Graph y = 2 sin 4x continued

Step 4
Plot the points and join them with a
smooth curve
0


8
4
3
8

2
Sine curveis invertedbecause " a" is negative
Amplitudeis | a |  | -2 |  2
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Slide 4-27
Homework

4.1 Page 141
All: 1 – 32

MyMathLab Assignment 4.1 for practice

MyMathLab Homework Quiz 4.1 will be due for a
grade on the date of our next class meeting

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Slide 4-28
4.2
Translations of the Graphs of the
Sine and Cosine Functions
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Horizontal Translations





The horizontal movement of a graph to the left or right of
its original position is called a “horizontal translation”
In circular functions, a horizontal translation is called a
phase shift.
A phase shift in a circular function occurs when a term is
added or subtracted from an argument:
When a positive number “d” is added to an argument the
phase shift is “d” units to the left
When a positive number “d” is subtracted from an
argument, the phase shift is “d” units to the right
Copyright © 2005 Pearson Education, Inc.
Slide 4-30
Example of a Phase Shift in the Sine
Function

Compared with y  sin x , the graph of



y  sin x   will be moved
units to
2
2

the left.



 Phaseshift willbe  
2

The amplitude will still be “1” and the period will
still be 2
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Slide 4-31
Sketch the graph of y  sin x   



2
The primary period of this function will occur

when: 0  x   2
2
Solving for x gives:
0  2 x    4
   2 x  3

3
  x 
2
2
Phase Shift
3   
Period:
     2
2  2
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Slide 4-32
Sketch the graph of y  sin x   




2
3
Divide this interval:   x 
into quarter
2
2
points:


3
 , 0,
, ,
2
2
2
Assign pattern of sine values (0, 1, 0, -1, 0) to
each of these five numbers:
  
 
 3 
  , 0 , 0, 1,  , 0 ,  ,  1,  , 0 
 2 
2 
 2 

Connect the points with a smooth curve:
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Slide 4-33
Sketch the graph of y  sin x   


2
Points to be connected with smooth curve:
  
 
 3 
  , 0 , 0, 1,  , 0 ,  ,  1,  , 0 
 2 
2 
 2 


y  sin x  
2

1


2 1
0

2

3
2
2
Note : Phase shift is
Copyright © 2005 Pearson Education, Inc.
y  sin x

2
units left
Slide 4-34
Graph y = sin (x  /3)

Find the interval for one primary period.
0 x

 2
3

7
x
3
3

Divide the interval into four equal parts.


5
4
11
7
,
,
,
,
3
6
3
6
3
Assign pattern of sine values (0, 1, 0, -1, 0) to each of
these five numbers:    5   4   11
  7

, 0 , 
,  1, 
, 0
 , 0 ,  ,1, 
3   6   3   6
  3 
Copyright © 2005 Pearson Education, Inc.
Slide 4-35
Graph y = sin (x  /3) continued

Plot:
    5   4   11
  7 
,
0
,
,
1
,
,
0
,
,

1
, 0

 
 
 
, 
3   6   3   6
  3 


y  sin x  
3


3
5
6
Note: Phaseshift
Copyright © 2005 Pearson Education, Inc.
4
3

3
11
6
7
3
y  sin x


 to right
3

Slide 4-36
Graph y  3cos  x   

4

Note: There will be both an amplitude change and a
phase shift.

0  x   2
Find the interval of primary period:

Divide into four equal parts.




4
,

4
,
3
,
4
5
,
4
4

7
 x
4
4
7
4
Assign pattern of cosine values, (1, 0, -1, 0, 1), multiplied
by 3, to each of these five numbers (3, 0, -3, 0, 3):
       3
  5   7 

,
3
,
,
0
,
,

3
, 3

 
 
,  , 0 , 
 4  4   4
  4   4 
Copyright © 2005 Pearson Education, Inc.
Slide 4-37


Graph y  3cos  x   continued
4


Plot:    , 3,   , 0 ,  3 ,  3,  5 , 0 ,  7 , 3
 4
 4
  4
  4
  4



y  3 cos x  
4


Copyright © 2005 Pearson Education, Inc.


4
4
3
4
5
4
7
4
π
Note: Phaseshift 4
All haveperiod2
y  3 cos x
y  cos x


 to left
4

Slide 4-38
Vertical Translations





The vertical movement of a graph up or down from its
original position is called a “vertical translation”
A vertical translation of a circular function occurs when a
term is added to, or subtracted from, a basic function
When a positive number “c” is added to a basic function
the vertical translation is “c” units up
When a positive number “c” is subtracted from a basic
function the vertical translation is “c” units down
In both these cases the effect is to move the “x-axis”
from y = 0 to y = c
Copyright © 2005 Pearson Education, Inc.
Slide 4-39
Example of a Vertical Translation in the Sine
Function



Compared with y  sin x , the graph of
y  1 sin x will be moved 1 unit up
The amplitude will still be “1” and the period will
still be 2
There will be no phase shift
Copyright © 2005 Pearson Education, Inc.
Slide 4-40
Sketch the graph of y  1 sin x

The primary period of this function will occur when:
0  x  2

Divide this interval into quarter points:
0,



3
, ,
, 2
2
2
Assign pattern of sine values (0, 1, 0, -1, 0), with “1”
added to each, to each of these five numbers:
 
 3 
0, 1,  , 2 ,  , 1,  , 0 , 2 , 1
2 
 2

Connect the points with a smooth curve:
Copyright © 2005 Pearson Education, Inc.
Slide 4-41
Sketch the graph of y  1 sin x

Points to be connected with smooth curve:
0, 1,   , 2 ,  , 1,  3 , 0 , 2 , 1
2

 2

In effect thewhole graph,including the x - axis, has been movedup
one unit and amplitudeis measuredrelativeto the" new x - axis"
2
y  1 sin x
1
1
0
Copyright © 2005 Pearson Education, Inc.

2

3
2
2
y  sin x
Slide 4-42
Analyzing Functions of the Form:
y  c  a sin b( x  d ) or y  c  a cosb( x  d )







Before analyzing transformations, these functions should
be written in the exact form shown here, including
having the argument factored as shown.
Compared with the basic functions: y  sin x or y  cos x
“a” changes the amplitude from 1 to | a |
if “a” is negative, the graph is inverted
2
2

to
“b” changes the period from
b
“c” causes a vertical translation of “c” units
“d” causes a phase shift (horizontal translation) of “d”
units (If “d” is positive, then “x – d” gives a phase shift “d”
units to the right, and “x + d” gives a phase shift “d” units
left)
Copyright © 2005 Pearson Education, Inc.
Slide 4-43
Analyze y = 2  2 sin 3x compared with
y = sin x




The graph will have an amplitude of 2 and will
be inverted
2
Its period will be
3
It will have a vertical translation of +2
(2 units up)
It will have no phase shift
Copyright © 2005 Pearson Education, Inc.
Slide 4-44
Further Guidelines for Sketching
Graphs of Sine and Cosine Functions




To graph y = c + a sin b(x - d) or y = c + a cos b(x – d),
with b > 0, follow these steps.
Step 1
To find the interval for one period, solve the
inequality: 0  bx  d   2 and lay
off this interval on the x-axis (the number at
the right end will be the new period)
Step 2
Find numbers to divide the interval into four
equal parts.
Step 3
Assign the appropriate sine (0, 1, 0, -1, 0) or
cosine (1, 0, -1, 0, 1) patterns, multiplied by
“a”, and with “c” added, to each of these five
x-values.
Copyright © 2005 Pearson Education, Inc.
Slide 4-45
Guidelines for Sketching Graphs of
Sine and Cosine Functions continued

Step 4

Step 5
Plot the points found in Step 3, and join
them with a smooth curve having
amplitude |a|.
Draw the graph over additional
periods, to the right and to the left, as
needed.
Copyright © 2005 Pearson Education, Inc.
Slide 4-46
Sketch the graph of y  2  2 sin 3x
0  3x  2
2
0 x
3
Endpointsand QuarterPoints:
0,

6
,

3
,

2
,
2
3
Sine pattern(0,1, 0, - 1, 0), multiplied
by - 2, with 2 added :
2, 0, 2, 4, 2
4
2
2
0
Copyright © 2005 Pearson Education, Inc.


6
3
2
2 3

Slide 4-47
Analyze y = 1 + 2 sin (4x + ) compared
with y = sin x


First factor the coefficient of x from argument:


Period:
y  1  2 sin 4 x  
4

2 

4
2


Phase Shift: 

Amplitude: 2

Vertical Translation:  1
Copyright © 2005 Pearson Education, Inc.
4
Slide 4-48
Graph y = 1 + 2 sin (4x + )
0  4 x    2
   4x  


4
x
Endpointsand QuarterPoints:



4
, 

8
, 0,

8
,

4
Sine pattern(0,1, 0, - 1, 0), multiplied
by 2, with - 1 added :
4
 1, 1,  1,  3,  1
1


4

 1
8 3
0
Copyright © 2005 Pearson Education, Inc.

8

4
Slide 4-49
Homework

4.2 Page 152
All: 1 – 12, 17 – 22, 27 – 28, 31 – 34, 39 – 46

MyMathLab Assignment 4.2 for practice

MyMathLab Homework Quiz 4.2 will be due for a
grade on the date of our next class meeting

Copyright © 2005 Pearson Education, Inc.
Slide 4-50
4.3
Graphs of Other Circular
Functions
Copyright © 2005 Pearson Education, Inc.
Graphs of Cosecant and Secant Functions




Since these functions are reciprocals of the sine and
cosine functions, they also have period 2
The graph of one period can be done over any interval of
the domain that has this length, but we will call the
interval 0, 2  the primary period
At the values of the domain where sine and cosine have
values of 0, cosecant and secant will be undefined,
these values of the domain establish vertical
asymptotes, shown on the graphs as vertical dashed
lines (asymptotes are not actually part of the graph, but
they show where a function is undefined)
Graphs will be continuous between these vertical
asymptotes
Copyright © 2005 Pearson Education, Inc.
Slide 4-52
Graphs of Cosecant and Secant Functions



You will recall that values of the cosecant and
secant functions (ranges) will be  , 1  1 , 
so no portion of their graphs will be in the
interval  1, 1
Cosecant and secant functions will have value 1
and -1 when sine and cosine functions have
these values
Other values of cosecant and secant functions
can be found by reciprocating sine and cosine
values
Copyright © 2005 Pearson Education, Inc.
Slide 4-53
Cosecant Function
y  sin x
Copyright © 2005 Pearson Education, Inc.
Slide 4-54
Secant Function
y  cos x
Copyright © 2005 Pearson Education, Inc.
Slide 4-55
Reference Graphs for Cosecant and Secant




By reciprocal identities already learned
1
1
csc bx  d  
and secbx  d  
sin bx  d 
cosbx  d 
To sketch the graphs of cosecant or secant with these
arguments, we can first sketch the corresponding sine
and cosine graphs with those same arguments as
references
Where the sine and cosine graphs have value of 0 draw
vertical asymptotes to show places where the cosecant
and secant functions are undefined
Reciprocate the values of the sine and cosine functions
and sketch the graphs of cosecant and secant
Copyright © 2005 Pearson Education, Inc.
Slide 4-56


Graph Primary Period of y  sec 3 x  
2



y  cos3 x  
2



0  3 x    2
2

 2
0 x 
2
3
0  6 x  3  4
3  6 x  7

7
x
2
6
Copyright © 2005 Pearson Education, Inc.
Endpointsand QuarterPoints:

2
,
2
,
3
5
, ,
6
7
6
Cosine pattern(1, 0, -1, 0,1)


Graph y  sec 3 x   by reciprocating
2

referencefunction
3
6
7
6
Slide 4-57
Graphing y  c  a csc bx  d 

First graph the reference function:

Note that functions boxed in red are reciprocals
Also note that both functions have undergone
the same vertical stretching or squeezing
and, the same vertical translation
Therefore, to complete the graph is it only
necessary to attached the U-shaped cosecant
curve to the reference curve



y  c  a sin bx  d 
Copyright © 2005 Pearson Education, Inc.
Slide 4-58
Graphing y  c  a sec bx  d 

First graph the reference function:

Note that functions boxed in red are reciprocals
Also note that both functions have undergone
the same vertical stretching or squeezing
and, the same vertical translation
Therefore, to complete the graph is it only
necessary to attached the U-shaped secant
curve to the reference curve



y  c  a cosbx  d 
Copyright © 2005 Pearson Education, Inc.
Slide 4-59
Sketch Graph: y  1  3 sec 1  x   
2
Copyright © 2005 Pearson Education, Inc.
Slide 4-60
Visualizing and Graphing the 2Tangent
Function
2
3
cos
2
1

3
2
1
3
2
tan
 2  3
1
3

2
2
tan
  1 .7
3
sin
2
3

3
2
Copyright © 2005 Pearson Education, Inc.
1
2
Slide 4-61
Observations About the Tangent
Function and Graph of y = tan x





The tangent will be undefined at
and every odd
2
multiple of it (the graph will have vertical asymptotes at
these values of the domain)
The graph will be continuous and all possible values
of
  
tangent will be obtained as x varies between   , 
 2 2
Based on this last observation, the period of the tangent
  
function will be  and   ,  will be called the primary
 2 2
period
Since there are no maximum or minimum values for
tangent, amplitude is not defined
Copyright © 2005 Pearson Education, Inc.
Slide 4-62
Tangent
Function
Endpoints &
Quarter Points:



2
, 

4
, 0,

4
,

2
Tangent
Values:

 ,  1, 0, 1, 
Copyright © 2005 Pearson Education, Inc.
Slide 4-63
Cotangent Function




This function is the reciprocal of the tangent
function
At places where the tangent has value of 0,
cotangent will be undefined and the graph will
have vertical asymptotes at those values

At odd multiples of
where the tangent is
2
undefined, the value of cotangent is 0
Using these facts and taking reciprocals of
tangent values the graph of cotangent can be
established as follows:
Copyright © 2005 Pearson Education, Inc.
Slide 4-64
Cotangent Function as a Reciprocal of
Tangent




The period is: 
Primary period: 0, 
Endpoints and Quarter Points:
Cotangent values:

0 
4
3 
4



3
0,
,
,
, 
4
2
4
, 1, 0,  1,  
y  cot x
y  tan x
Copyright © 2005 Pearson Education, Inc.
Slide 4-65
Cotangent Function
Copyright © 2005 Pearson Education, Inc.
Slide 4-66
Sketching Graphs of Transformed
Tangent Functions


To graph y  c  a tanbx  d 
Determine the primary period by solving for x:





2
 b x  d  

2
Find endpoints and quarter points
Assign pattern of tangent values  , 1, 0, 1, 
multiplied by “a” with “c” added
Note: Analysis of phase shift, period change,
vertical stretching or squeezing, and vertical
translation will be the same as for sine and
cosine
Copyright © 2005 Pearson Education, Inc.
Slide 4-67
Sketch
Copyright © 2005 Pearson Education, Inc.
y  1  3 tan2 x   
Slide 4-68
Sketching Graphs of Transformed
Cotangent Functions





To graph y  c  a cot bx  d 
Determine the primary period by solving for x:
0  bx  d   
Find endpoints and quarter points
Assign pattern of tangent values , 1, 0, 1,  
multiplied by “a” with “c” added
Note: Analysis of phase shift, period change,
vertical stretching or squeezing, and vertical
translation will be the same as for sine and
cosine
Copyright © 2005 Pearson Education, Inc.
Slide 4-69
Sketch
1

y  1  3 cot  x  
2
4
Copyright © 2005 Pearson Education, Inc.
Slide 4-70
Homework

4.3 Page 165
All: 1 – 6,
Even: 8 – 46

MyMathLab Assignment 4.3 for practice

MyMathLab Homework Quiz 4.3 will be due for a
grade on the date of our next class meeting
Copyright © 2005 Pearson Education, Inc.
Slide 4-71
4.4
Harmonic Motion
Copyright © 2005 Pearson Education, Inc.
Simple Harmonic Motion


The position of a point oscillating about an
equilibrium position at time t is modeled by
either s(t )  a cos t
or s(t )  a sin t
where a and  are constants, with   0. The
amplitude of the motion is |a|, the period is 2

and the frequency is  .
2
Copyright © 2005 Pearson Education, Inc.
Slide 4-73
Example




Suppose that an object is attached to a coiled
spring such the one shown (on the next slide). It
is pulled down a distance of 5 in. from its
equilibrium position, and then released. The time
for one complete oscillation is 4 sec.
a) Give an equation that models the position of
the object at time t.
b) Determine the position at t = 1.5 sec.
c) Find the frequency.
Copyright © 2005 Pearson Education, Inc.
Slide 4-74
Example continued


When the object is
released at t = 0, distance
the object of the object
from its equilibrium
position 5 in. below
equilibrium.
We use s(t )  cos t


a = 5
2

 4,
or  
Copyright © 2005 Pearson Education, Inc.

2
Slide 4-75
Example continued

The motion is modeled by 5cos  t.
2

b) After 1.5 sec, the position is


s (1.5)  5cos  (1.5)   2.54 in.
2


Since 3.54 > 0, the object is above the
equilibrium position.
c) The frequency is the reciprocal of the period,
of ¼.
Copyright © 2005 Pearson Education, Inc.
Slide 4-76